2010-02-06 08:18:29 UTC
February 6, 2010
This Week's Finds in Mathematical Physics (Week 293)
This week I want to list a bunch of papers and books on n-categories.
Then I'll tell you about a conference on the math of environmental
sustainability and green technology. And then I'll continue my story
about electrical circuits. But first...
This column started with some vague dreams about n-categories and
physics. Thanks to a lot of smart youngsters - and a few smart
oldsters - these dreams are now well on their way to becoming reality.
They don't need my help anymore! I need to find some new dreams. So,
"week300" will be the last issue of This Week's Finds in Mathematical
I still like learning things by explaining them. When I start work at
the Centre for Quantum Technologies this summer, I'll want to tell you
about that. And I've realized that our little planet needs my help a
lot more than the beautiful structure of the universe does! The deep
secrets of math and physics are endlessly engrossing - but they can
wait, and other things can't. So, I'm trying to learn more about ecology,
economics, and technology. And I'd like to talk more about those.
So, I plan to start a new column. Not completely new, just a bit
different from this. I'll call it This Week's Finds, and drop the
"in Mathematical Physics". That should be sufficiently vague that I
can talk about whatever I want.
I'll make some changes in format, too. For example, I won't keep
writing each issue in ASCII and putting it on the usenet newsgroups.
Sorry, but that's too much work.
I also want to start a new blog, since the n-Category Cafe is not the
optimal place for talking about things like the melting of Arctic ice.
But I don't know what to call this new blog - or where it should
reside. Any suggestions?
I may still talk about fancy math and physics now and then. Or even
a lot. We'll see. But if you want to learn about n-categories, you
don't need me anymore. There's a *lot* to read these days. I mentioned
Carlos Simpson's book in "week291" - that's one good place to start.
Here's another introduction:
1) John Baez and Peter May, Towards Higher Categories, Springer, 2009.
Also available at http://ncatlab.org/johnbaez/show/Towards+Higher+Categories
This has a bunch of papers in it, namely:
* John Baez and Michael Shulman, Lectures on n-categories and cohomology.
* Julia Bergner, A survey of (infinity,1)-categories.
* Simona Paoli, Internal categorical structures in homotopical algebra.
* Stephen Lack, A 2-categories companion.
* Lawrence Breen, Notes on 1- and 2-gerbes.
* Ross Street, An Australian conspectus of higher categories.
After browsing these, you should probably start studying
(infinity,1)-categories, which are infinity-categories where all the
n-morphisms for n > 1 are invertible. There are a few different
approaches, but luckily they're nicely connected by some results
described in Julia Bergner's paper. Two of the most important
approaches are "Segal spaces" and "quasicategories". For the latter,
2) Andre Joyal, The Theory of Quasicategories and Its Applications,
and then go here:
3) Jacob Lurie, Higher Topos Theory, Princeton U. Press, 2009.
Also available at http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf
This book is 925 pages long! Luckily, Lurie writes well. After
setting up the machinery, he went on to use (infinity,1)-categories
to revolutionize algebraic geometry:
4) Jacob Lurie, Derived algebraic geometry I: stable infinity-categories,
available as arXiv:math/0608228.
Derived algebraic geometry II: noncommutative algebra, available as
Derived algebraic geometry III: commutative algebra, available as
Derived algebraic geometry IV: deformation theory, available as
Derived algebraic geometry V: structured spaces, available as
Derived algebraic geometry VI: E_k algebras, available as
For related work, try these:
5) David Ben-Zvi, John Francis and David Nadler, Integral transforms
and Drinfeld centers in derived algebraic geometry available as
6) David Ben-Zvi and David Nadler, The character theory of a complex
group, available as arXiv:0904.1247.
Lurie is now using (infinity,n)-categories to study topological
quantum field theory. He's making precise and proving some old
guesses James Dolan and I had:
7) Jacob Lurie, On the classification of topological field theories,
available as arXiv:0905.0465.
Jonathan Woolf is doing it in a somewhat different way, which I hope
will be unified with Lurie's work eventually:
8) Jonathan Woolf, Transversal homotopy theory, available as
All this stuff is starting to transform math in amazing ways. And I
hope physics, too - though so far, it's mainly helping us understand
the physics we already have.
Meanwhile, I've been trying to figure out something else to do. Like
a lot of academics who think about beautiful abstractions and soar
happily from one conference to another, I'm always feeling a bit
guilty, wondering what I could do to help "save the planet". Yes, we
recycle and turn off the lights when we're not in the room. If we all
do just a little bit... a little will get done. But surely mathematicians
have the skills to do more!
I'm sure lots of you have had such thoughts. That's probably why Rachel
Levy ran this conference last weekend:
9) Conference on the Mathematics of Environmental Sustainability and
Green Technology, Harvey Mudd College, Claremont, California,
Friday-Saturday, January 29-30, 2010. Organized by Rachel Levy.
Here's a quick brain dump of what I learned.
First, Harry Atwater of Caltech gave a talk on photovoltaic solar
10) Atwater Research Group, http://daedalus.caltech.edu/
The efficiency of silicon crystal solar cells peaked out at 24% in
2000. Fancy "multijunctions" get up to 40% and are still improving.
But they use fancy materials like gallium arsenide, gallium indium
phosphate, and so on. The world currently uses 13 terawatts of power.
The US uses 3. But building just 1 terawatt of these fancy
photovoltaics would use up more rare substances than we can get our
11) Gordon B. Haxel, James B. Hedrick, and Greta J. Orris, Rare earth
elements - critical resources for high technology, US Geological Survey
Fact Sheet 087-02, available at http://pubs.usgs.gov/fs/2002/fs087-02/
So, if we want solar power, we need to keep thinking about silicon and
use as many tricks as possible to boost its efficiency.
There are some limits. In 1961, Shockley and Quiesser wrote a paper
on the limiting efficiency of a solar cell. It's limited by
thermodynamical reasons! Since anything that can absorb energy
can also emit it, any solar cell also acts as a light-emitting diode,
turning electric power back into light:
12) W. Shockley and H. J. Queisser, Detailed balance limit of
efficiency of p-n junction solar cells, J. Appl. Phys. 32 (1961)
13) Wikipedia, Schockley-Quiesser limit,
What are the tricks used to approach this theoretical efficiency?
Multijunctions use layers of different materials to catch photons of
different frequencies. The materials are expensive, so people use a
lens to focus more sunlight on the photovoltaic cell. The same is true
even for silicon - see the Umuwa Solar Power Station in Australia.
But then the cells get hot and need to be cooled.
Roughening the surface of a solar cell promotes light trapping, by
large factors! Light bounces around ergodically and has more chances
to get absorbed and turned into useful power. There are theoretical
limits on how well this trick works. But those limits were derived
using ray optics, where we assume light moves in straight lines. So,
we can beat those limits by leaving the regime where the ray-optics
approximation holds good. In other words, make the surface
complicated at length scales comparable to the wavelength at light.
For example: we can grow silicon wires from vapor! They can form
densely packed structures that absorb more light:
14) B. M. Kayes, H. A. Atwater, and N. S. Lewis, Comparison of the
device physics principles of planar and radial p-n junction nanorod
solar cells, J. Appl. Phys. 97 (2005), 114302.
Also, with such structures the charge carriers don't need to travel
so far to get from the n-type material to the p-type material. This
also boosts efficiency.
There are other tricks, still just under development. Using quasiparticles
called "surface plasmons" we can adjust the dispersion relations to
create materials with really low group velocity. Slow light has more
time to get absorbed! We can also create "meta-materials" whose
refractive index is really wacky - like n = -5!
I should explain this a bit, in case you don't understand. Remember,
the refractive index of a substance is the inverse of the speed of
light in that substance - in units where the speed of light in vacuum
equals 1. When light passes from material 1 to material 2, it takes
the path of least time - at least in the ray-optics approximation.
Using this you can show Snell's law:
sin(theta_1)/sin(theta_2) = n_2/n_1
where n_i is the index of refraction in the ith material and theta_i
is the angle between the light's path and the line normal to the
interface between materials.
Air has an index of refraction close to 1. Glass has an index of
refraction greater than 1. So, when light passes from light to glass,
it "straightens out": its path becomes closer to perpendicular to the
air-glass interface. When light passes from glass to air, the reverse
happens: the light bends more. But the sine of an angle can never exceed
1 - so sometimes Snell's law has no solution. Then the light gets
stuck! More precisely, it's forced to bounce back into the glass.
This is called "total internal reflection", and the easiest way to see
it is not with glass, but water. Dive into a swimming pool and look
up from below. You'll only see the sky in a limited disk. Outside
that, you'll see total internal reflection.
Okay, that's stuff everyone learns in optics. But *negative* indices
of refraction are much weirder! The light entering such a material
will bend *backwards*.
Materials with a negative index of refraction also exhibit a reversed
version of the ordinary Goos-Hanchen effect. In the ordinary version,
light "slips" a little before reflecting during total internal
reflection. The "slip" is actually a slight displacement of the
light's wave crests from their expected location - a "phase slip".
But for a material of negative refractive index, the light slips
*backwards*. This allows for resonant states where light gets
trapped in thin films. Maybe this can be used to make better solar
Next, Kenneth Golden gave a talk on sea ice, which covers 7-10% of the
ocean's surface and is a great detector of global warming. He's a
mathematician at the University of Utah who also does measurements in
the Arctic and Antarctic. If you want to go to math grad school
without becoming a nerd - if you want to brave 70-foot swells, dig
trenches in the snow and see Emperor penguins - you want Golden as
15) Ken Golden's website, http://www.math.utah.edu/~golden/
Salt gets incorporated into sea ice via millimeter-scale brine
inclusions between ice platelets, forming a "dendritic platelet
structure". Melting sea ice forms fresh water in melt ponds atop the
ice, while the brine sinks down to form "bottom water" driving the
global thermohaline conveyor belt. You've heard of the Gulf Stream,
right? Well, that's just part of this story.
When it gets hotter, the Earth's poles get less white, so they absorb
more light, making it hotter: this is "ice albedo feedback". Ice
albedo feedback is *largely controlled by melt ponds*. So if you're
interested in climate change, questions like the following become
important: when do melt ponds get larger, and when do they drain out?
Sea ice is diminishing rapidly in the Arctic - much faster than all
the existing climate models had predicted. There's a lot less sea ice
in the Antarctic, mainly in the Wedell Sea, and there it seems to be
growing, maybe due to increased precipitation. In the Arctic, winter
sea ice diminished in area by about 10% from 1978 to 2008. But summer
sea ice diminished by about 40%! It took a huge plunge in 2007,
leading to a big increase in solar heat input due to the ice albedo
16) Donald K. Perovich, Jacqueline A. Richter-Menge, Kathleen
F. Jones, and Bonnie Light, Sunlight, water, and ice: Extreme Arctic
sea ice melt during the summer of 2007, Geophysical Research Letters,
35 (2008), L11501. Also available at
There's a lot of interesting math involved in understanding the
dynamics of sea ice. The ice thickness distribution equation was
worked out by Thorndike et al in 1975. The heat equation for ice and
snow was worked out by Maykut and Understeiner in 1971. Sea ice
dynamics was studied by Kibler.
Ice floes have two fractal regimes, one from 1 to 20 meters, another
from 100 to 1500 meters. Brine channels have a fractal
character well modeled by "diffusion limited aggregation". Brine
starts flowing when there's about 5% of brine in the ice - a kind of
percolation problem familiar in statistical mechanics. Here's what it
looks like when there's 5.7% brine:
17) Kenneth Golden, Brine inclusions in a crystal of lab-grown sea ice,
Nobody knows why polycrystalline metals have a log-normal distribution
of crystal sizes. Similar behavior, also unexplained, is seen in sea
A "polynya" is an area of open water surrounded by sea ice. Polynyas
occupy just .001% of the overall area in Antarctic sea ice, but create
1% of the icea. Icy cold catabatic winds blow off the mainland,
pushing away ice and creating patches of open water which then refreeze.
There was anomalous export of sea ice through Fran Strait in the 1990s,
which may have been one of the preconditions for high ice albedo feedback.
20-40% of sea ice is formed by surface flooding followed by refreezing.
This was *not included* in the sea ice models that gave such
The food chain is founded on diatoms. These form "extracellular
polymeric substances"- goopy mucus-like stuff made of polysaccharides
that protects them and serves as antifreeze. There's a lot of this
stuff; the ice gets visibly stained by it.
For more, see:
18) Kenneth M. Golden, Climate change and the mathematics of transport
in sea ice, AMS Notices, May 2009. Also available at
19) Mathematics Awareness Month, April 2009: Mathematics and Climate,
Next, Julie Lundquist, who just moved from Lawrence Livermore Labs
to the University of Colorado, spoke about wind power:
20) Julie Lunquist, Department of Atmospheric and Oceanic Sciences,
University of Colorado, http://paos.colorado.edu/people/lundquist.php
With increased reliance on wind, the power grid will need to be
redesigned to handle fluctuating power sources. In the US, currently,
companies aren't paid for power they generate in excess of the amount
they promised to make. So, accurate prediction is a hugely important
game. Being off by 1% can cost millions of dollars! Europe has
different laws, which encourage firms to maximize the amount of wind
power they generate.
If you had your choice about where to build a wind turbine, you'd
build it on the ocean or a very flat plain, where the air flows rather
smoothly. Hilly terrain leads to annoying turbulence - but sometimes
that's your only choice. Then you need to find the best spots, where
the turbulence is least bad. Complete simulation of the Navier-Stokes
equations is too computationally intensive, so people use fancier tricks.
There's a lot of math and physics here.
For weather reports people use "mesoscale simulation" which cleverly
treats smaller-scale features in an averaged way - but we need more
fine-grained simulations to see how much wind a turbine will get. This
is where "large eddy simulation" comes in.
A famous Brookhaven study suggested that the power spectrum of wind
has peaks at 4 days, 1/2 day, and 1 minute. This perhaps justifies an
approach where different time scales, and thus length scales, are
treated separately and the results then combined somehow. The study
is actually a bit controversial. But anyway, this is the approach
people are taking, and it seems to work.
Night air is stable - but day air is often not, since the ground is
hot, and hot air rises. So when a parcel of air moving along hits a
hill, it can just shoot upwards, and not come back down! This means
lots of turbulence.
Eddy diffusivity is modeled by Monin-Obukhov similarity theory:
21) American Meteorological Society Glossary, Monin-Obukhov similarity theory,
The wind turbines at Altamont Pass in California kill more raptors
than all other wind farms in the world combined! Old-fashioned wind
turbines look like nice places to perch, spelling death to birds.
Cracks in concrete attract rodents, which attract raptors, who get
killed. The new ones are far better.
22) National Renewable Energy Laboratory, Research needs for winds
resource characterization, available as
Finally, there was a talk by Ron Lloyd of Fat Spaniel Technologies.
This is a company that makes software for solar plants and other
sustainable energy companies:
23) Fat Spaniel Technologies, http://www.fatspaniel.com/products/
His talk was less technical so I didn't take detailed notes. One big
point I took away was this: we need better tools for modelling! This
is especially true with the coming of the "smart grid". In its
simplest form, this is a power grid that uses lots of data - for
example, data about power generation and consumption - to regulate
itself and increase efficiency. Surely there will be a lot of math
here. Maybe even the topic I've been talking about lately: bond graphs!
But now I want to talk about some very simple aspects of electrical
circuits. Last week I listed various kinds of circuits. Now let's go
into a bit more detail - starting with the simplest kind: circuits
made of just wires and linear resistors, where the currents and
voltages are independent of time.
Mathematically, such a circuit is a graph equipped with some extra data.
First, each edge has a number associated to it - the "resistance". For
| | |
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2 3 2
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Second, we have current flowing through this circuit. To describe this,
we first arbitrarily pick an orientation on each edge:
| | |
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V V V
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Then we label each edge with a number saying how much "current"
is flowing through that edge, in the direction of the arrow:
| | |
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3V V1 V 3
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| | |
Electrical engineers call the current I. Mathematically it's good
to think of I as a "1-chain": a linear combination of oriented edges
of our graph, with the coefficients of the linear combination being
the numbers shown above.
If we know the current, we can work out a number for each vertex of
our graph, saying how much current is flowing out of that vertex,
minus how much is flowing in:
1 o---->----o---->----o 0
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| | |
V V V
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-5 o----<----o---->----o 0
Mathematically we can think of this as a "0-chain": a formal linear
combination of the vertices of our graph, with the numbers shown above
as coefficients. We call this 0-chain the "boundary" of the 1-chain
we started with. Since our current was called I, we call its boundary
Kirchhoff's current law says that
delta I = 0
When this holds, let's say our circuit is a "closed". Physically this
follows from the law of conservation of electrical charge, together
with a reasonable assumption. Current is the flow of charge. If the
total current flowing into a vertex wasn't equal to the amount flowing
out, charge - positive or negative - would be building up there. But
for a closed circuit, we assume it's not.
If a circuit is not closed, let's call it "open". These are interesting
too. For example, we might have a circuit like this:
where we have current flowing in the wire on top and flowing out the
two wires at bottom. We allow delta I to be nonzero at the ends of
these wires - the 3 vertices labelled x. This circuit is an "open
system" in the sense of "week290", because it has these wires dangling
out of it. It's not self-contained; we can use it as part of some
bigger circuit. We should really formalize this more, but I won't
now. Derek Wise did it more generally here:
24) Derek Wise, Lattice p-form electromagnetism and chain field theory,
available as gr-qc/0510033.
The idea here was to get a category where chain complexes are morphisms
in a category. In our situation, composing morphisms amounts to gluing
the output wires of one circuit into the input wires of another. This
is an example of the general philosophy I'm trying to pursue, where
open systems are treated as morphisms.
We've talked about 1-chains and 0-chains... but we can also back up
and talk about 2-chains! Let's suppose our graph is connected - it is
in our example - and let's fill it in with enough 2-dimensional
"faces" to get something contractible. We can do this in a god-given
way if our graph is drawn on the plane: just fill in all the holes!
In electrical engineering these faces are
often called "meshes".
This give us a chain complex
C_0 <-------- C_1 <-------- C_2
and a cochain complex:
C^0 -------> C^1 -------> C^2
As I've already said, it's good to think of the current I as a 1-chain,
delta I = 0
is Kirchoff's current law. Since our little space is contractible the
above equation implies that
I = delta J
for some 2-chain J called the "mesh current". This assigns to each
face or "mesh" the current flowing around that face.
An electrical circuit also comes with a third piece of data, which I
haven't mentioned yet. Each oriented edge should be labelled by a
number called the "voltage" across that edge. Electrical engineers
call the voltage V. It's good to think of V as a 1-cochain, which
assigns to each edge the voltage across that edge.
Why a 1-cochain instead of a 1-chain? Because then
d V = 0
is the other basic law of electrical circuits - Kirchhoff's voltage
law! This law says that the sum of these voltages around a mesh is
zero. Since our little space is contractible the above equation
V = d phi
for some 0-cochain phi called the "electrostatic potential". In
electrostatics, this potential is a function on space. Here it
assigns a number to each vertex of our graph.
Since the space of 1-cochains is the dual of the space of 1-chains, we
can take the voltage V and the current I, glom them together, and get
This the "power": that is, the rate at which our network soaks up
energy and dissipates it into heat. Note that this is just a fancy
version of formula for power that I explained in "week290" - power is
effort times flow.
I've given you three basic pieces of data labelling our circuit: the
resistance R, the current I, and the voltage V. But these aren't
independent! Ohm's law says that the voltage across any edge is the
current through that times the resistance of that edge. But this
remember: voltage is a 1-cochain while current is a 1-chain. So
"resistance" can be thought of as a map from 1-cochains to 1-chains:
R: C^1 -> C_1
This lets us write Ohm's law like this:
V = RI
This, in turn, means the power of our circuit is
V(I) = (RI)(I)
For physical reasons, this power is always nonnegative. In fact,
let's assume it's positive unless I = 0. This is just another way of
saying that resistance labelling each edge is positive. It can be
very interesting to think about circuits with perfectly conducting
wires. These would give edges whose resistance is zero. But that's a
bit of an idealization, and right now I'd rather allow only *positive*
Why? Because then we can think of the above formula as the inner
product of I with itself! In other words, then there's a unique inner
product on 1-cochains with
(RI)(I) = <I,I>
In this situation
R: C^1 -> C_1
is the usual isomorphism that we get between a finite-dimensional
inner product space and its dual. (For this statement to be true,
we'd better assume our graph has finitely many vertices and edges.)
Now, if you've studied de Rham cohomlogy, all this should start
reminding you of Hodge theory. And indeed, it's a baby version
of that! So, we're getting a little bit of Hodge theory, but in
a setting where our chain complexes are really morphisms in a category.
There's a lot more to say, but I want dinner.
Quote of the Week:
"So many young people are forced to specialize in one line or another that
a young person can't afford to try and cover this waterfront - only an old
fogy who can afford to make a fool of himself. If I don't, who will?" -
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
For a table of contents of all the issues of This Week's Finds, try
A simple jumping-off point to the old issues is available at
If you just want the latest issue, go to